towards a general theory for coupling functions allowing ... tiago.pereira@ ... synchronisation...

Towards a general theory for coupling functions

allowing persistent synchronisation

Tiago Pereira1, Jaap Eldering1, Martin Rasmussen1, and Alexei

Veneziani2

1Department of Mathematics, Imperial College London, London SW7 2AZ, UK2Centro de Matematica, Computacao e Cognicao, UFABC, Santo Andre, Brazil

E-mail: [email protected], [email protected],

[email protected]

Abstract.

We study synchronisation properties of networks of coupled dynamical systems with

interaction akin to diffusion. We assume that the isolated node dynamics possesses a

forward invariant set on which it has a bounded Jacobian, then we characterise a class

of coupling functions that allows for uniformly stable synchronisation in connected

complex networks — in the sense that there is an open neighbourhood of the initial

conditions that is uniformly attracted towards synchronisation. Moreover, this stable

synchronisation persists under perturbations to non-identical node dynamics. We

illustrate the theory with numerical examples and conclude with a discussion on

embedding these results in a more general framework of spectral dichotomies.

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Coupling functions allowing persistent synchronisation 2

1. Introduction

Network synchronisation is observed to occur in a broad range of applications in

physics [33], neuroscience [6, 12, 30, 20], and ecology [8]. During the last fifty years,

empirical studies of real complex systems have led to a deep understanding of the

structure of networks [21, 2], and the interaction properties between oscillators, that

is, the coupling function [18, 31, 34].

The stability of network synchronisation is a balance between the isolated dynamics

and the coupling function. Past research suggests that in networks of identical oscillators

with interaction akin to diffusion, under mild conditions on the isolated dynamics, the

coupling function dictates the synchronisation properties of the network [24, 19, 25, 23,

34]. However, it still remains an open problem to describe the class of coupling functions

that lead the network to persistent synchronisation.

Our work contributes to the development a general theory for coupling functions

that allow for persistent synchronisation for a connected complex network. The coupling

functions under consideration appear in a variety of synchronisation models on networks

(such as the Kuramoto models [18] and its extensions [1, 5, 27]).

More precisely, we consider the dynamics of a network of n identical elements with

interaction akin to diffusion, described by

xi = f(t, xi) + αn∑

j=1

Wijh(xj − xi) , (1)

where α is the overall coupling strength, and the matrix W = (Wij)i,j∈{1,...,n} describes

the interaction structure of the network, i.e. Wij measures the strength of interaction

between the nodes i and j. The function f : R × Rm → Rm describes the isolated

node dynamics, and the coupling function h : Rm → Rm describes the diffusion-like

interaction between nodes. We make the following two assumptions for these functions.

Assumption A1. The function f is continuous, and there exists an inflowing invariant

open ball U ⊂ Rm such that f is continuously differentiable in U with

‖D2f(t, x)‖ ≤ % for all t ∈ R and x ∈ Ufor some % > 0.

For instance, the Lorenz system has a bounded inflowing invariant ball, see

Subsection 3.2. In general, smooth nonlinear systems with compact attractors satisfy

Assumption A1. This assumption will be generalised in Section 5 to include also

noncompact sets U .

Assumption A2. The coupling function h is continuously differentiable with h(0) = 0.

We define Γ := Dh(0) and denote the (complex) eigenvalues of Γ by βi, i ∈ {1, . . . ,m}.The network structure plays a central role for the synchronisation properties. We

consider the intensity of the i-th node Vi =∑n

j=1 Wij, and define the positive definite

matrix V := diag(V1, . . . , Vn). Then the so-called Laplacian reads as

L = V −W .


Let λi, i ∈ {1, . . . , n}, denote the eigenvalues of L. Note that λ1 = 0 is an eigenvalue

with eigenvector 1√n(1, . . . , 1). The multiplicity of this eigenvalue equals the number of

connected components of the network.

The following assumption incorporates the coupling and structural network

properties.

Assumption A3. We suppose that

γ := min2≤i≤n1≤j≤m

Re(λiβj) > 0 ,

where Re(z) denotes the real part of a complex number z.

The dynamics of such a diffusive model can be intricate. Indeed, even if the isolated

dynamics possesses a globally stable fixed point, the diffusive coupling can lead to

instability of the fixed point and the system can exhibit an oscillatory behaviour [28].

Note that due to the diffusive nature of the coupling, if all oscillators start with the

same initial condition, then the coupling term vanishes identically. This ensures that

the globally synchronised state x1(t) = x2(t) = . . . = xn(t) = s(t) is an invariant state

for all coupling strengths α and all choices of coupling functions h. That is, the diagonal

manifold

M := {xi ∈ Rm for i ∈ {1, · · · , n} : x1 = · · · = xn}is invariant, and we call the subset

S := {xi ∈ U ⊂ Rm for i ∈ {1, · · · , n} : x1 = · · · = xn} ⊂M (2)

the synchronisation manifold. The main result of this paper is a proof that under the

general conditions given above and α sufficiently large, the synchronisation manifold S

is uniformly exponentially stable.

Theorem 1 (synchronisation). Consider the network of diffusively coupled equations (1)

satisfying A1–A3. Then there exists a ρ = ρ(f,Γ) such that for all coupling strengths

α >ρ

γ,

the network is locally uniformly synchronised. This means that there exist a δ > 0

and a C = C(L,Γ) > 0 such that if xi(t0) ∈ U and ‖xi(t0) − xj(t0)‖ ≤ δ for any

i, j ∈ {1, . . . , n}, then

‖xi(t)− xj(t)‖ ≤ Ce−(αγ−ρ)(t−t0)‖xi(t0)− xj(t0)‖ for all t ≥ t0 . (3)

Hence, the synchronisation manifold is locally uniformly exponentially attractive.

The constant ρ depends on the bounds on the Jacobian D2f as set out in Assumption A1

and on the conditional number of the matrix Γ (see (27) in case Γ is diagonalisable). In

the case that the Laplacian L and Γ are diagonalisable, C depends on the conditional

number of the similarity transformation that diagonalises these matrices (see Lemma 8

for details), so loosely speaking, it depends on how well the eigenvectors of L and Γ are

orthogonal. If L and Γ are non-diagonalisable, then C is related to conditional numbers


as well, see the proof of Lemma 9 for details. The size of δ can be estimated explicitly

if more concrete details about the system are known, see also Remark 15 on page 22.

Our second main result shows that synchronisation is persistent under perturbation

of the isolated nodes. Thereto, consider a network of non-identical nodes described by

xi = fi(t, xi) + αn∑

j=1

Wijh(xj − xi), (4)

where fi(t, xi) = f(t, xi) + gi(t, xi). Note that in this case, the synchronisation manifold

S is no longer invariant. We show in this paper that for small perturbations functions

gi, i ∈ {1, . . . , n}, the synchronisation manifold is stable in the sense that orbits starting

near the synchronisation manifold S remain in a neighbourhood of S.

Theorem 2 (persistence). Consider the perturbed network (4) of diffusively coupled

equations fulfilling Assumptions A1–A3, and suppose that

α >ρ

γ

as in Theorem 1. Then there exist δ > 0, C > 0 and εg > 0 such that for all ε0-

perturbations satisfying

‖gi(t, x)‖ ≤ ε0 ≤ εg for all t ∈ R , x ∈ U and i ∈ {1, . . . , n} (5)

and initial conditions satisfying ‖xi(t0)− xj(t0)‖ ≤ δ for any i, j ∈ {1, . . . , n}, the

estimate

‖xi(t)− xj(t)‖ ≤ Ce−(αγ−ρ)(t−t0)‖xi(t0)− xj(t0)‖+Cε0

αγ − ρ for all t ≥ t0 (6)

holds.

Note that the additional term Cε0/(αγ−ρ) can be made small either by controlling

the perturbation size ε0 or by increasing αγ. This provides control of the network

coherence in terms of the network properties and coupling strength.

If the Laplacian L is symmetric (i.e. the systems are mutually coupled), its spectrum

is real and can be ordered as 0 = λ1 < λ2 ≤ λ3 ≤ . . . ≤ λn. Moreover, consider

β := mini∈{1,...,m}Reβi, and note that this implies

γ = βλ2 .

The following corollary to the above persistence result then shows that the

enhancement of coherence in the network in terms of network connectivity depends

on the spectral gap λ2.

Corollary 3 (synchronisation error). Consider the perturbed network (4) with

symmetric Laplacian L and the average synchronisation error

es(t) =1

n(n− 1)

n∑

i,j=1

‖xi(t)− xj(t)‖ for all t ≥ t0 ,


where the initial conditions xi(t0), i ∈ {1, . . . , n}, are chosen as in Theorem 2. Then

whenever αγ = αβλ2 > ρ, one has

lim supt→∞

es(t) ≤ Kε0

αβλ2 − ρ,

where K = K(Γ) is independent of the network size.

This corollary has excellent agreement with recent numerical simulations for

the synchronisation transition in complex networks of mutually coupled non-identical

oscillators [26].

The paper is organised as follows. In Section 2, we discuss our assumptions, ideas

of the proofs as well as how our results relates to previous contributions. In Section 3,

we illustrate our main synchronisation result with a nonautonomous linear system and

a coupled Lorenz system. Section 4 provides fundamental results on nonautonomous

linear differential equations. In Section 5, we provide auxiliary results to prove our main

theorems in Sections 6 and 7. Finally, in Section 8, we discuss how to generalise this

theory using the dichotomy spectrum and normal hyperbolicity.

Notation. We endow the vector space Rm with the Euclidean norm ‖x‖ =√∑m

i=1 |xi|2and the associated Euclidean inner product. In addition, we equip the vector space

(Rm)n = Rnm with the norm

‖(x1, . . . , xn)‖ := maxi=1,...,n

‖xi‖ where xi ∈ Rm . (7)

Note that linear operators on the above spaces will be equipped with the induced

operator norm. For a given invertible matrix A ∈ Rd×d, the conditional number is

defined by κ(A) = ‖A‖‖A−1‖. Note that the conditional number depends on the

underlying operator norm. Finally, the symbol Id stands for the identity matrix in

Rd.

2. Discussion of the main results

This section is devoted to relating our results to the state of the art and to explaining

the assumptions and the central ideas of the proofs.

2.1. State of the art

Recent research on synchronisation has focused on the role of the coupling function

for the stability of network synchronisation. Notably, Pecora and collaborators

have developed so-called master stability functions to estimate Lyapunov exponents

corresponding to the transversal directions of the synchronisation manifold [24, 15]. In

contrast to this approach, we estimate the contraction rate by dichotomy techniques.

Our results show that the synchronisation state is locally stable and persistent, and

thus stable under small perturbations. This means that the phenomenon of bubbling [3]

and riddling [13] (which leads to synchronisation loss) will not be observed under our

conditions, in contrast to the master stability function approach.


Another aspect of our results is that the synchronisation properties do not depend

on diagonalisation properties of the Laplacian. Recently, the master stability function

has been extended to include non-diagonalisable Laplacians [22]. However, these results

do not guarantee that an open neighbourhood of the synchronisation manifold will

be attracted by the synchronisation manifold, nor do they imply persistence of the

synchronisation. In our set-up, these properties follow naturally by means of roughness

of exponential dichotomies, which is relevant in applications that are subjected to noise

and external influences. Note that the master stability function approach is applicable

to a broader class of coupling functions than the ones we consider, but our approach is

constructive and making use of further dichotomy techniques and normal hyperbolicity

our results can be generalised further, as discussed later in Section 8.

In addition, Pogromsky and Nijmeijer [29] use control techniques to show that if

the coupling function is linear and given by a symmetric positive definite matrix, then

the synchronisation manifold is globally asymptotically stable for connected networks.

Likewise, Belykh, Belykh and Hasler [4] develop a connection graph stability method to

obtain global synchronisation for the network, by assuming the existence of a quadratic

Lyapunov function associated with the isolated system. In this article, we tackle only

local stability properties, but we consider a more general class of coupling functions.

However, under additional conditions on the dynamics and coupling functions, it is

possible to prove global stability with the techniques we have developed by applying the

mean value theorem instead of using Taylor expansions of the vector field.

2.2. The assumptions

Our main assumptions are natural and fulfilled by a large class of systems. Assumption

A1 concerns the existence of solutions and the boundedness of the Jacobian. Assumption

A2 makes it possible to characterise the stability of synchronisation by the linearisation

of h. Assumption A3 guarantees that the eigenvalues of the tensor L⊗Γ have real part

bounded away from zero (except for the trivial eigenvalue).

These hypotheses basically imply that with a finite value of α, we are able to damp

all the instabilities of the vector field and obtain a stable synchronisation state. If for

example, Assumption A3 is dropped, γ may become negative and synchronisation may

no longer be possible.

We illustrate the relevance of Assumption A3 with the following example. Consider

the isolated dynamics f : R2 → R2 given by f(x) = −εx. Moreover, consider three

coupled systems

x1 = f(x1) + 2αΓ(x2 − x1) + αΓ(x3 − x1),

x2 = f(x2) + 2αΓ(x3 − x2)

x3 = f(x3) + αΓ(x1 − x3)


with

Γ =

(2 1

−17 0

)and note that L =

3 −2 −1

0 2 −2

−1 0 1

.

The eigenvalues of L are λ1 = 0, λ2 = 3 + i and λ3 = 3− i and the eigenvalues of Γ are

β1 = 1 + 4i and β2 = 1− 4i. Hence,

γ = −1,

and although the isolated dynamics has a stable trivial fixed point, for any α > ε the

origin is unstable and there are trajectories of the coupled systems that escape any

compact set. This shows that breaking condition A3 can have severe effects on the

dynamics of the coupled systems.

Assumption A3 has not been considered in the literature to our best knowledge. In

the following, we rephrase this condition in the following two special cases:

(i) The spectrum of Γ is positive. If Γ has a spectrum consisting of only real, positive

eigenvalues, then A3 has a representation in terms of the Laplacian. In this case,

this condition reads as

Re(λi) > 0 for all i 6= 1 ,

since the Laplacian always has a zero eigenvalue. If the network is connected,

this eigenvalue is simple, and by virtue of the disk theorem, a sufficient condition

for all other eigenvalues to have positive real part is positive interaction strength,

i.e. Wij > 0 whenever i is connected to j, and zero otherwise.

(ii) The Laplacian is symmetric. This is the most studied case in the literature.

Assume that the network is connected. Since the spectrum of the Laplacian is

real, Assumption A3 requires that the real part of the spectrum of Γ is positive and

that the spectrum of the Laplacian is positive apart from the single zero eigenvalue

(or alternatively, that the spectra of Γ and the Laplacian are both negative, but

note that this is non-physical).

2.3. Ideas of the proofs

The proofs of our main results rely on identifying the synchronisation problem with a

corresponding fixed point problem. We first concentrate on the case of diagonalisable

Laplacians, where diagonal dominance (Proposition 6) can be used to show that

the synchronised state is uniformly asymptotically stable. To obtain the claim for

general coupling functions, we make use of the roughness property associated with the

equilibrium point (Theorem 5). The main aspect here is to approximate the coupling

function by a diagonalisable one while keeping control of the contraction rates. Finally,

the proof for general Laplacians follows from the fact that the set of diagonalisable

Laplacians is dense in the space of Laplacians. From these results and the roughness

property the main claim follows.


3. Illustrations

Before proving the two main results of this paper, two examples are discussed.

3.1. Nonautonomous Linear Equations

Consider the nonautonomous linear equation

x = A(t)x (8)

where

A(t) =

(−1− 9 cos2(6t) + 12 sin(6t) cos(6t) 12 cos2(6t) + 9 sin(6t) cos(6t)

−12 sin2(6t) + 9 sin(t) cos(6t) −1− 9 sin2(6t)− 12 sin(6t) cos(6t)

).

This is a prototypical example where the eigenvalues of the time-dependent matrices do

not characterise the stability of a nonautonomous linear system. Indeed, the eigenvalues

of A(t) are −1 and −10, independent of t ∈ R, and a direct computation shows that

x(t) =

(e2t(cos(6t) + 2 sin(6t)) + 2e−13t(2 cos(6t)− sin(6t))

e2t(cos(6t)− 2 sin(6t)) + 2e−13t(2 cos(6t)− sin(6t))

)

is a solution of the system, which does not converge to 0 as t→∞.

Consider now two diffusively coupled systems

x1 = A(t)x1 + αΓ(x2 − x1) ,

x2 = A(t)x2 + αΓ(x1 − x2) ,

where Γ is a real 2× 2 matrix. Theorem 1 yields that it is possible to synchronise these

two systems for any coupling matrix with β(Γ) > 0. Consider the coupling matrix

Γ =

(β 1

0 β

).

Γ is in its Jordan form and non-diagonalisable. The transformation y = x1 − x2 leads

to

y = (A(t)− 2αΓ)y . (9)

Our main result shows that the trivial solution of (9) is stable if α is large enough.

We have integrated (9) using a sixth order Runge–Kutta method with step size

0.001. We have computed the critical coupling value αc as a function of β, such that

the trivial solution of Eq. (9) is stable. In Figure (1) we plotted the corresponding

critical value ρc = βαc. Hence, we are able to analyse the dependence of ρ on f and Γ.

The behaviour of ρ appears to be intricate. For large β, we obtain that ρ tends to a

constant, however, as we decrease β, various changes in the behaviour can be observed.

Although the problem is linear, the critical coupling strength depends nonlinearly on

the parameter β. We analyse this dependence in more details in Section 6.1


0.01 0.1 1 10!

1

100

! cρ

Figure 1. ρ = ρ(f,Γ) as a function of β in a log–log scale for a fixed f given by

Eq. (8). For small β the slope is −1 in good approximation.

3.2. The Lorenz system

Using the notation x = (u, v, w), the Lorenz vector field is given by

f(x) =

σ(v − u)

u(r − w)− v−bw + uv

,

where we choose the classical parameter values σ = 10, r = 28 and b = 83. All trajectories

of the Lorenz system enter a compact set eventually and exist globally forward in time for

this reason. Moreover, they accumulate in a neighbourhood of a chaotic attractor [32].

Consider the network of three coupled Lorenz systems

xi = f(xi) + α3∑

j=1

WijH(xj − xi) , (10)

where the interaction matrix W is given as in Figure 2.

W =

0 1 11 0 aa 1 0

Figure 2. The network and its weight matrix. The matrix L = V − W is non-

diagonalisable for every a 6= 1; here we choose a = 13 .


We use two different nonlinear coupling functions; for the first, the associated matrix

Γ is positive definite, whereas for the second, Γ is a Jordan block. The specific forms of

the coupling functions can be seen in Figure 3. We have integrated (10) using a sixth

order Runge–Kutta method with step size 0.0001 and computed the critical coupling αcas a function of β, and then plotted the value ρc = αcβ (see Figure 3). The behaviour of

ρ depends in an essential way on Γ. This behaviour is further discussed in Section 6.1.

0,1 1 10

1

100

c

0,1 1 10

!0

0,5

1

1,5

2

!c

Γ =

β 0 00 β 00 0 β

Γ =

β 1 00 β 10 0 β

h(x) =

βu + vβ sin v + wβw(1 − u)

ρρ

h(x) =

βu + w2

uv + β sin vβw(1 − u)

Figure 3. Simulation results for ρ for the two coupling functions. For the first case, see

left side, Γ = βI is positive definite for β > 0, and the behaviour of ρ does not depend

significantly on β. For the second case, Γ is a Jordan block with eigenvalues equal to

β. In this situation, for large values of β, the critical coupling ρ appears independent

of β, as opposed to the small values of β. In that case, the critical coupling scales as

ρ ∝ β−1.

4. Nonautonomous linear differential equations

Consider the m-dimensional linear differential equation

x = A(t)x (11)

where x ∈ Rm and A : R→ Rm×m is a bounded and continuous matrix function. Recall

that solutions of (11) can be written in terms of the evolution operator Φ : R × R →Rm×m; the solution for the initial condition x(t0) = x0 is given by

t 7→ Φ(t, t0)x0 .

Definition 4 (uniform exponential stability). Consider the linear system (11) with

evolution operator Φ. System (11) is said to be uniformly exponentially stable if there

exists K,µ > 0 such that

‖Φ(t, t0)‖ ≤ Ke−µ(t−t0) for all t ≥ t0 . (12)


The following roughness theorem guarantees that uniform exponential stability is

persistent under perturbations. A proof can be found in [7, Lecture 4, Prop. 1].

Theorem 5 (roughness). Consider the linear system (11) and assume that for K > 0

and µ ∈ R, the evolution operator Φ satisfies the exponential estimate

‖Φ(t, t0)‖ ≤ Ke−µ(t−t0) for all t ≥ t0 . (13)

Consider a continuous matrix function V : R→ Rm×m such that

δ := supt∈R‖V (t)‖ <∞ .

Then the evolution operator Φ of the perturbed equation

y = (A(t) + V (t))y

satisfies the exponential estimate

‖Φ(t, t0)‖ ≤ Ke−µ(t−t0) for all t ≥ t0 ,

where µ := µ− δK.

There are various criteria to obtain conditions for uniform exponential stability.

We shall use the following criterion for diagonal dominant matrices, which can be found

in [7, Lecture 6, Prop. 3].

Proposition 6 (diagonal dominance criterion). Consider the linear system (11) with

complex time-dependent coefficient matrices A(t) = (Aij(t))i,j=1,...,m, and suppose that

there exists a constant µ > 0 such that

Re(Aii(t))+m∑

j=1,j 6=i

|Aij(t)| ≤ −µ < 0 for all t ∈ R and i ∈ {1, . . . ,m} .(14)

Then the evolution operator Φ of (11) satisfies

‖Φ(t, t0)‖ ≤ Ke−µ(t−t0) for all t ≥ t0 .

with K = K(m) ≥ 1.

5. Auxiliary results

In this section, we obtain various exponential estimates for orbits near the

synchronisation manifold S of (1). First, we introduce a convenient splitting of

coordinates along the synchronisation manifold and complementary to it, and derive

the equations with respect to these coordinates. Then we prove linear stability of

the synchronisation manifold. Here we distinguish between diagonalisable and non-

diagonalisable Laplacians. The latter case will follow from approximation results on

diagonalisable Laplacians and roughness of the exponential estimates. Finally, we

introduce the concept of a tubular neighbourhood as a final ingredient to tackle the

general proof of nonlinear stability.


In order to treat noncompact absorbing sets U in Assumption A1, we reformulate

this assumption as follows.

Assumption A1’. The function f is continuous in the first argument and continuously

differentiable in the second argument, and there exists an open simply connected set

U ⊂ Rm with C1-boundary that is ε-inflowing invariant for some ε > 0, i.e. for all

x ∈ ∂U with inward-pointing normal vector qx, we have

〈qx, f(t, x)〉 ≥ ε for all t ∈ R and x ∈ ∂U . (15)

Moreover, there exists a ∆ > 0 such that the Jacobian D2f is uniformly continuous and

bounded on B∆(U) :=⋃x∈U{y ∈ Rm : ‖x− y‖ < ∆}, i.e. for some % > 0, we have

‖D2f(t, x)‖ ≤ % for all t ∈ R and x ∈ B∆(U) .

Note that if the closure U is compact, then uniformity of the inflowing invariance

condition as well as the uniform continuity of D2f and existence of a bound % follow

automatically. In the noncompact case, we require uniform bounds on the ∆-enlarged

neighbourhood B∆(U) for technical reasons.

We first obtain equations that govern the dynamics near the synchronisation

manifold. Using a tensor representation, we can write the nm-dimensional system (1)

equations by means of a single equation. To this end, define

X := col(x1, . . . , xn) ,

where col denotes the vectorisation formed by stacking the column vectors xi into a

single column vector. Similarly, define

F (t,X) := col(f(t, x1), . . . , f(t, xn)) .

We can analyse small perturbations away from the synchronisation manifold in terms

of the tensor representation

X = 1⊗ s+ ξ , (16)

where ⊗ is the tensor product and 1 = col(1, . . . , 1) ∈ Rn, which is the eigenvector of

L corresponding to the eigenvalue zero. Note that 1⊗ s defines the diagonal manifold,

and we view ξ as a perturbation to the synchronised state.

The state space Rn⊗Rm can be canonically identified with Rnm, which we will use

for shorter notation. The coordinate splitting (16) is associated to a splitting of Rnm as

the direct sum of subspaces

Rnm = M ⊕Nwith associated projections

πM : Rnm →M, πN : Rnm → N.

The subspaces M,N ⊂ Rnm are determined by embeddings from Rm and R(n−1)m,

respectively, induced by the Laplacian L on Rn.


Let us for the moment use the simplifying assumption that L is diagonalisable with

eigenvectors 1, v2, . . . , vn. Then the subspaces M,N have natural representations in

terms of these eigenvectors as

M = span(1)⊗ Rm , N = span(v2, . . . , vn)⊗ Rm .

This means that we have ‘natural’ embeddings that induce coordinates on these

subspaces:

ιM : Rm →M , s 7→ 1⊗ s = col(s, . . . , s) ,

ιN : R(n−1)m → N , (y2, . . . , yn) 7→n∑

j=2

vj ⊗ yj .

If we drop the assumption that L is diagonalisable, then we lose the natural choice of

an embedding for N . Note, however, that N is still determined as the eigenspace of all

non-zero eigenvalues.

Note that the norm on Rnm we chose is the maximum over the Euclidean norm on

Rm, see (7). The norm ‖·‖ on Rnm can be restricted to the subspaces M,N and induces

norms on the ‘coordinate’ spaces Rm and R(n−1)m by pullback under the embeddings.

Then the induced norm on s ∈ Rm is given by

‖s‖ιM = ‖ιM(s)‖ = ‖1⊗ s‖ , (17)

which is precisely the Euclidean norm. Similarly, ιM induces an inner product on M .

Henceforth, we shall identify s ∈ Rm with 1⊗ s ∈M under the isometry ιM .

Using the representation (16) for X ∈ Rnm, given an initial condition X0 = (s0, ξ0),

the corresponding solution to (1) reads as X(t) = (s(t), ξ(t)). In the next result,

we derive differential equations for these two components in a neighbourhood of the

synchronisation manifold.

Proposition 7. The two components of the solution X(t) = (s(t), ξ(t)) satisfy the

system of equations

1⊗ s = 1⊗ f(t, s) +Rs(s, ξ) , (18)

ξ = T (t, s)ξ +Rξ(s, ξ) , (19)

where

T (t, s) = In ⊗D2f(t, s)− α(L⊗ Γ) (20)

and R∗ := Rs, Rξ are the remainder functions such that for any ε > 0, there is a δ > 0

such that for all ‖ξ‖ ≤ δ, one has ‖R∗(s, ξ)‖ ≤ ε‖ξ‖.

Proof. By Assumption A2, Taylor’s theorem implies that given ε > 0, there exists a

δ > 0 such that

h(x) = Γ x+ r(x) with ‖r(x)‖ ≤ ε‖x‖ whenever ‖x‖ ≤ δ .


Now we define

Rh(X)i =n∑

j=1

Wijr(xi − xj) =n∑

j=1

Wijr(pi(1⊗ s+ ξ)− pj(1⊗ s+ ξ))

=n∑

j=1

Wijr(pi(ξ)− pj(ξ)) ,

where pi : Rnm → Rm maps canonically to the i-th component of the argument,

i ∈ {1, . . . , n}. The vectors Rh(X)i ∈ Rm, i ∈ {1, . . . , n} define a vector in Rnm.

Note that Rh(X) = Rh(ξ) does not depend on s ∈M and satisfies the estimate

‖Rh(ξ)‖ ≤ maxi=1,...,n

( n∑

j=1

|Wij|)ε 2‖ξ‖ whenever ‖ξ‖ ≤ δ

2.

Recall that Lij = δijVi −Wij, so the coupling term can then be rewritten asn∑

j=1

Wijh(xj − xi) = −n∑

j=1

LijΓxj +Rh(ξ)i (21)

The Taylor expansion of F (t,X) around 1⊗ s reads as

F (t,1⊗ s+ ξ) = F (t,1⊗ s) +D2F (t,1⊗ s)ξ +RF (t, s, ξ)

= 1⊗ f(t, s) + In ⊗D2f(t, s)ξ +RF (t, s, ξ),

where ‖RF (t, s, ξ)‖ ≤ ε‖ξ‖ when ‖ξ‖ ≤ δ. An algebraic manipulation of (21) allows a

representation in coordinates (s, ξ) ∈M ⊕N of the n equations forming (1):

X = 1⊗ s+ ξ = 1⊗ f(t, s) + In ⊗D2f(t, s)ξ − α(L⊗ Γ)ξ

+RF (t, s, ξ) + αRh(ξ), (22)

where we used L1 = 0. Hence, the term (L⊗ Γ)(1⊗ s) vanishes.

Next, we project the differential equation (22) onto the spaces M and N to obtain

differential equations for s and ξ:

1⊗ s = 1⊗ f(t, s) + πM(RF (t, s, ξ) + αRh(ξ)),

ξ = T (t, s)ξ + πN(RF (t, s, ξ) + αRh(ξ)),

where

T (t, s) = In ⊗D2f(t, s)− α(L⊗ Γ).

Note that both In ⊗ D2f(t, s) and L ⊗ Γ preserve the subspaces M and N , since Inand L preserve both span(1) and span(v2, . . . , vn), so the projections can be dropped

there.


5.1. Diagonalisable Laplacians

We now prove stability of the linear flow (20) for ξ ∈ N , along any curve s(t) ∈ S,

which is not necessarily a solution. We first treat the diagonalisable case, and then the

non-diagonalisable one. Then, in Section 6, we use these results to prove stability of the

fully nonlinear problem.

Lemma 8 (Diagonalisable case). Consider the linearisation of (19), given by

ξ = T (t, s(t))ξ , ξ ∈ N (23)

with s(t) ∈ U , and the representations

L = PΛP−1 and Γ = QBQ−1

with P ∈ Rn×n and Q ∈ Rm×m, such that Λ = diag(λ1, λ2, . . . , λn) and B =

diag(β1, . . . , βm). Then there exists a ρ > 0 such that for all coupling strengths

α >ρ

γ,

the evolution operator Φ of (23) satisfies the estimate

‖Φ(t, t0)‖ ≤ Kκ(P ⊗Q) e−(αγ−ρ)(t−t0) for all t ≥ t0 ,

with K ≥ 1, and where κ(P ⊗Q) denotes the conditional number of P ⊗Q.

Note that for matrices P ∈ Rn×n and Q ∈ Rm×m, we obtain

‖P ⊗Q‖ ≤ ‖P‖∞‖Q‖2 ,

which implies that κ(P ⊗Q) ≤ κ∞(P )κ2(Q).

Proof of Lemma 8. Note that O := P⊗Q is an invertible matrix that diagonalises L⊗Γ,

and the change of coordinates

T (t) = O−1 T (t, s(t))O = In ⊗Q−1D2f(t, s(t))Q− αΛ⊗B (24)

reduces T (t) to m-block diagonal form. Thus, we have

T (t) =n⊕

i=1

Ti(t) = diag(T1(t), . . . , Tn(t)) ,

where

Ti(t) := Q−1D2f(t, s(t))Q︸︷︷︸A(t):=

−αλiB for all t ∈ R .

Since for all t ∈ R, the matrix T (t) is block diagonal, the dynamics given by Y = T (t)Y

preserves the splitting Rnm =⊕n

i=1 Rm, and hence, its associated evolution operator Φ

is also of the form

Φ(t, t0) =n⊕

i=1

Φi(t, t0) for all t, t0 ∈ R , (25)


where each Φi is the evolution operator of yi = Ti(t)yi. Note that restricting T to

N corresponds to restricting T to the blocks i ≥ 2. The dynamics in each block is

determined by

yi = (A(t)− αλiB)yi . (26)

Now define

ρ := supt∈R, s∈U

‖A(t)‖ .

Note that the matrix A(t) depends implicitly on s(t) ∈ U , so by Assumption A1 we get

the estimate

ρ ≤ κ(Q)% . (27)

To apply Proposition 6, we search for a condition on α such that

Re(Akk − αλiβk) +∑

1≤j≤mj 6=k

|Akj(t)| < 0 for all k ∈ {1, . . . ,m} . (28)

Since Re(Akk) ≤ |Akk|, it is therefore sufficient that

α >

∑mj=1 |Akj|

Re(λiβk)

holds. Note that Re(λiβk) ≥ γ, so if we definem∑

j=1

|Aij| ≤ cρ =: ρ ,

where c > 0 depends on the choice of the norm. Then by the diagonal dominance

criterion (Proposition 6), the evolution operator Φi satisfies

‖Φi(t, t0)‖ ≤ Ke−(αγ−ρ)(t−t0) for all t ≥ t0 . (29)

Finally, using (25) and changing back to the original coordinates, we have

‖Φ(t, t0)‖ = ‖O(⊕

i≥2 Φi(t, t0))O−1‖≤ κ(O) maxi≥2 ‖Φi(t, t0)‖≤ Kκ(O) e−(αγ−ρ)(t−t0) for all t ≥ t0 . (30)

Note that O−1 maps M and N onto the first and last n − 1 of the m-tuples in Rnm

respectively, so the restriction to N reduces to a direct sum over i ≥ 2 after conjugation

with O, while we can simply estimate κ(O|O−1N) ≤ κ(O).


5.2. Non-diagonalisable Laplacian

We now treat the case when the Laplacian is non-diagonalisable and Γ is diagonalisable.

Note that if Γ is non-diagonalisable, the results follow from the density of diagonalisable

matrices and the roughness property.

Lemma 9 (Non-diagonalisable Laplacian). Consider the situation of Lemma 8 without

the condition that the Laplacian is diagonalisable. Then there exists a ρ > 0 such that

for all coupling strengths

α >ρ

γ,

the evolution operator Φ of (23) satisfies the estimate

‖Φ(t, t0)‖ ≤ Ce−(αγ−ρ)(t−t0) for all t ≥ t0 ,

where C = C(Γ, L) ≥ 1.

The proof of this lemma makes use of roughness of exponential dichotomies and the

density of diagonalisable Laplacians. We first establish the following auxiliary result.

Proposition 10. Let ε > 0 and J be a complex Jordan block of dimension m. Consider

J = J + E

where E = diag(0, ε, 2ε, . . . , (m−1)ε). Then there exists an R ∈ Rm×m such that R−1JR

is diagonal and

‖R−1ER‖ = bε

with the constant b = b(m).

Proof. Note that J is diagonalisable, since all the eigenvalues are distinct. All

corresponding transformations R are matrices of eigenvectors, upper triangular and can

be computed explicitly. We normalise the eigenvectors such that for `, j ∈ {1, . . . ,m}

R`j :=

{(j−1)!(j−`)!ε

`−1 for all ` ≤ j ,

0 otherwise

It is easy to verify that the elements R−1ik with i, k ∈ {1, . . . ,m} of the inverse of R read

as

R−1ik =

{(−1)i+k

(i−1)!(k−i)!ε−(k−1) for all i ≤ k ,

0 otherwise

We have

(R−1ER)ij =∑

k,`

R−1ik Ek`R`j = ε

(−1)i(j − 1)!

(i− 1)!

j∑

k=i

(−1)k(k − 1)

(j − k)!(k − i)! .


Note that (R−1ER)ii = (i − 1)ε. Likewise, we have (R−1ER)i,i+1 = −iε. Moreover, if

j > i+ 1 then (R−1ER)ij = 0, since

j∑

k=i

(−1)k(k − 1)

(j − k)!(k − i)! = (−1)ij−i∑

l=1

(−1)l

(l − 1)!(j − i− l)! = 0 .

Therefore, max1≤i 6=m∑m

j=1 |(R−1ER)ij| = max{(2m − 3),m − 1}ε, and the result

follows.

Now we are ready to prove our approximation result.

Proposition 11. Let L be a Laplacian with simple eigenvalue zero and 1 its associated

eigenvector. Then for any ε > 0, there exists a matrix L with simple eigenvalue zero

and 1 its associated eigenvector such that

(i) L = P ΛP−1 with a diagonal matrix Λ ∈ Rn×n, and

(ii) ‖P−1(L− L)P‖ ≤ ε.

Proof. We only need to prove the statement if L is non-diagonalisable. We decompose

L in its complex Jordan canonical form

L = OJO−1 ,

where J is a block diagonal matrix. The first block corresponds to the simple eigenvalue

zero, so the first row contains only zeros, that is, J = diag(0, J1, . . . , Jk), where Ji are

Jordan blocks corresponding to non-zero eigenvalues. Without loss of generality, we

consider k = 1.

Define v := O−11. By hypothesis, we have L1 = 0, so

Jv = 0 . (31)

As each Jordan block has its own invariant subspace, (31) implies v = (1, 0, . . . , 0).

Define E := diag(0, ε, 2ε, . . . , (n− 1)ε), and note that

Ev = 0 . (32)

Consider the matrix

L = O(J + E)O−1 ,

which is diagonalisable. Moreover, by (31) and (32), we obtain that L has zero as a

simple eigenvalue with associated eigenvector 1. By Proposition 10, we obtain

J + E = RΛR−1 ,

and hence the matrix P = OR diagonalises L. For this reason,

P−1(L− L)P = P−1(OEO−1)P = R−1ER ,

and the result follows by Proposition 10.


Proof of Lemma 9. As in the diagonalisable case, we consider the linearised

equation (23) for ξ ∈ N along any curve s(t) ∈ U . By Proposition 11, there is

a diagonalisable matrix L in an arbitrary neighbourhood of the Laplacian L. We

rewrite (23) as

ξ = [In ⊗D2f(t, s(t))− αL⊗ Γ]ξ + α[(L− L)⊗ Γ]ξ . (33)

Note that this is a small perturbation of the same equation with diagonalisable Laplacian

L, so we can apply the results from Subsection 5.1. Recall that Γ = QBQ−1

and L = P ΛP−1 (see Proposition 11). Moreover, consider the change of variables

ζ = (P−1 ⊗Q−1)ξ. We obtain

ζ = [In ⊗Q−1D2f(t, s(t))Q− αΛ⊗B]ζ + α[P−1(L− L)P ⊗B]ζ . (34)

We treat α(P−1(L− L)P ⊗B)ζ as a perturbation of the equation

ζ = (In ⊗Q−1D2f(t, s(t))Q− α(Λ⊗B))ζ . (35)

It follows from the proof of Lemma 8 (see (29) for details) that the evolution operator

Φ of (35) satisfies

‖Φ(t, t0)‖ ≤ Ke−(αγ−ρ)(t−t0) ,

where K does not depend on n as (35) is block diagonal. Theorem 5 (the roughness

theorem) implies that the condition

α‖P−1(L− L)P ⊗B‖ < αγ − ρK

(36)

leads to an exponential stability estimate for the perturbed equation (33). By

Proposition 11 (ii), we can choose L such that ‖P−1(L − L)P‖ ≤ ε/‖B‖, so (36) is

satisfied if taking ε < (αγ−ρ)/(αK). Hence, setting ρ := ρ+αKε, then for all α > ρ/γ

the linear flow Φ(t, t0) for (33) satisfies

‖Φ(t, t0)‖ ≤ Kκ(P ⊗Q)e−(αγ−ρ)(t−t0) for all t ≥ t0 ,

where the conditional number is due to transforming back to the original variables ξ.

To analyse the solution curves (s(t), ξ(t)) of the nonlinear system (18,19) we

introduce the concept of a tubular neighbourhood.

Definition 12 (η-tubular neighbourhood). Let S = 1 ⊗ U ⊂ M be a subset of the

diagonal manifold. Then the set

Sη = {1⊗ s+ ξ : s ∈ U and ξ ∈ N, where ‖ξ‖ < η} (37)

for a given η > 0 is called the η-tubular neighbourhood of S.

See Figure 4 for a schematic illustration of this definition. Note that the directions

along N in which the tubular stretches out do not need to be orthogonal to M .

Assumption A1’ says that the single-node system has a uniformly inflowing invariant

set U ⊂ Rm. A similar result holds in a neighbourhood of the synchronisation manifold

S in the coupled network, since the following lemma implies that if the solution curve

(s(t), ξ(t)) leaves Sη, then it must do so by ‖ξ(t)‖ growing larger than η.


η

Rm

Rm

Sη

∂cylSη

M

∂sideSη

Figure 4. Tubular neighbourhood Sη for n = 2.

Lemma 13. Consider Assumption A1’ with the ε-inflowing invariant set U ⊂ Rm.

Let X = F (t,X) describe the dynamics of n uncoupled copies of this system and let

G : R× Rnm → Rnm be a perturbation such that for some r > 0 and δ > 0, one has

supt∈R,X∈Sr

‖G(t,X)‖ ≤ δ <ε

‖πM‖.

Then there exists an η ∈ (0, r] such that solution curves (s(t), ξ(t)) of X = F (t,X) +

G(t,X) can only leave the tubular neighbourhood Sη through

∂cylSη := {1⊗ s+ ξ : s ∈ U and ‖ξ‖ = η} .

Proof. Choose η such that 0 < η ≤ r. The boundary of Sη consists of two parts:

∂Sη = ∂cylSη ∪ ∂sideSη ,

where ∂sideSη := {1⊗ s+ ξ : ‖ξ‖ ≤ η and s ∈ ∂U}.We consider the dynamics on ∂sideSη. Let q be the inward pointing normal vector

at s ∈ ∂U . Locally we have ∂sideSη = ∂S ⊕ N , so F + G points inwards at 1 ⊗ q + ξ

precisely if its projection onto M along N has positive inner product with q. Note that

we use the isometry ιM from (17) to endow M with the inner product 〈 · , · 〉M induced

from 〈 · , · 〉Rm , but no inner product on Rnm is used (nor defined).

If η is chosen sufficiently small, then Sη is contained within the product space

B∆(U)n where we have uniform bounds ‖D2F‖ ≤ % and ‖G‖ ≤ δ. It follows that

〈ιM(q), πM [F (t,1⊗ s+ ξ) +G(t,1⊗ s+ ξ)]〉M= 〈q, f(t, s)〉Rm + 〈q, ι−1

M ◦ πM [D2F (t,1⊗ s+ τ ξ)ξ +G(t,1⊗ s+ ξ)]〉Rm

≥ ε− ‖πM‖(‖D2F‖‖ξ‖+ ‖G‖)≥ ε− ‖πM‖(%η + δ),

where we applied the mean value theorem with τ ∈ (0, 1) as interpolation variable.

Since ‖πM‖δ < ε, there exists an η > 0 such that F + G points inwards everywhere at

∂sideSη.


Finally, we shall make use of the following lemma, which is a variant on Gronwall’s

Lemma.

Lemma 14. Let x(t) ∈ R satisfy the integral inequality

x(t) ≤ Ce−µ(t−t0)x0 +

∫ t

t0

Ce−µ(t−τ)(αx(τ) + β) dτ , (38)

with C, µ > 0 and x0, α, β ≥ 0, whenever x ≤ δ.

If µ := µ− Cα > 0 and x0 <1C

(δ − βµ), then x(t) is bounded by

x(t) ≤ Ce−µ(t−t0)(x0 −

β

µ

)+Cβ

µfor all t ≥ t0 , (39)

and in particular x(t) < δ holds for all t ≥ t0.

Proof. The integral inequality is equivalent to the differential inequality

x(t) ≤ −µx(t) + C(αx(t) + β) , x(t0) = Cx0 ,

so by a standard application of Gronwall’s lemma we obtain (39), as long as the solution

satisfies x(t) ≤ δ. Now assume by contradiction that this assumption is violated. Then

there exists a t1 ≥ t0 such that x(t) = δ for the first time at t = t1. However,

the assumption x(t) ≤ δ is true up to time t1, so by the previous estimates and the

assumption that x0 <1C

(δ − βµ) it follows that x(t1) < δ. This contradiction completes

the proof.

6. Synchronisation

In the previous section we have established all auxiliary results to prove our main

theorem on synchronisation (Theorem 1), which will be restated for convenience.

Theorem (synchronisation). Consider the network of diffusively coupled equations (1)

satisfying A1–A3. Then there exists a ρ = ρ(f,Γ) such that for all coupling strengths

α >ρ

γ,

the network is locally uniformly synchronised. This means that there exist a δ > 0

and a C = C(L,Γ) > 0 such that if xi(t0) ∈ U and ‖xi(t0) − xj(t0)‖ ≤ δ for any

i, j ∈ {1, . . . , n}, then

‖xi(t)− xj(t)‖ ≤ Ce−(αγ−ρ)(t−t0)‖xi(t0)− xj(t0)‖ for all t ≥ t0 .

Proof. Set

X(t0) = 1⊗ s(t0) + ξ(t0) := col(x1(t0), · · · , xn(t0))

where xi(t0) ∈ U and U ⊂ Rm is ε-inflowing invariant. Due to the uniformity

assumptions in A1’, there exists a slightly enlarged neighbourhood B∆/2(U) ⊃ U that

is still ε/2-inflowing invariant. We set S = 1 ⊗ B∆/2(U). If we choose the distance

bound ‖xi(t0)− xj(t0)‖ ≤ δ sufficiently small (depending on the angle between M and

N), then s(t0) ∈ S holds, while we also have ‖ξ(t0)‖ ≤ ‖πN‖δ.


By Lemma 13 there exists a tubular neighbourhood Sη of positive size η > 0 over

S that is inflowing invariant on the ‘side’ and contained within B∆(U)n ⊂ Rnm, so the

uniform assumptions of A1’ hold.

Now lemmas 8 and 9 together imply that there exists a ρ > 0 such that for α > ργ,

the evolution operator Φ(t, t0) for ξ satisfies an exponential estimate with decay rate

−(αγ − ρ). The nonlinear remainder of the flow of ξ can be bounded by an arbitrarily

small linear term when ‖ξ‖ is small, as controlled by η. By variation of constants,

Eq. (19) for ξ is equivalent to

ξ(t) = Φ(t, t0)ξ(t0) +

∫ t

t0

Φ(t, τ)Rξ(s(τ), ξ(τ)) dτ . (40)

Now we assume that ‖ξ(t)‖ ≤ η for all t ≥ t0 and estimate

‖ξ(t)‖ ≤ Ce−(αγ−ρ)(t−t0)‖πN‖δ +

∫ t

t0

Ce−(αγ−ρ)(t−τ)ε(η) dτ .

Hence, when we choose δ < ηC‖πN‖

and ε(η) sufficiently small, then we can apply

Lemma 14 with β = 0 and conclude that

‖ξ(t)‖ ≤ Ce−µ(t−t0)‖πN‖δ for all t ≥ t0 ,

with µ = αγ − ρ − Cε(η). Thus, if we choose ρ = ρ + Cε(η), then for all α > ργ

the

complete solution curve (s(t), ξ(t)) for the nonlinear system is contained in Sη for all

t ≥ t0 and converges to the synchronisation manifold S with decay rate −(αγ− ρ). The

explicit estimate for ‖xi(t)− xj(t)‖ can be recovered from

‖xi(t)− xj(t)‖ ≤ 2‖xi(t)− s(t)‖ ≤ 2‖ξ(t)‖and the fact that δ can be chosen smaller to match ‖xi(t)− xj(t)‖.

Remark 15. Explicit estimates for the size of δ in Theorem 1 can be found when more

details of the system are known. For example, if the second derivative of f is bounded,

i.e.∥∥D2

2f(t, x)∥∥ ≤ σ for all t ∈ R and x ∈ U ,

and the coupling function is linear, i.e. h(x) = Γx, then δ can be estimated as

δ =αγ − ρ

4σC‖πN‖. (41)

Note that for convenience, we ignore effects on the size of δ introduced by estimates at the

boundary of the synchronisation manifold. Under these assumptions the remainder Rξ

in (40) consists of RF , the nonlinearities of f , and can be estimated as ‖RF (t, s, ξ)‖ ≤σ‖ξ‖2 using mean value theorem arguments. To conclude the argument, fix δ =

η/(2C‖πN‖) and follow the proof of Theorem 1.


6.1. Behaviour of ρ as function of Γ

Our approach is constructive and allows to estimate the bounds for ρ = ρ(f,Γ) whenever

specific information on the function h is provided. By Lemma 9, it is clear that the

diagonalisation properties of the Laplacian have no effect on the bounds for ρ. In the

following, we only discuss symmetric Laplacians L. As an illustration, we look at two

cases for Γ.

(i) Γ is symmetric. There exists an orthogonal matrix Q such that Γ = QBQ−1.

Note that κ(Q) = 1 (i.e. the conditional number with respect to the Euclidean

norm). From (27), it follows that

ρ ≤ c%

for some c > 0. The bound for ρ is independent of Γ for this reason. Note that this

can be observed in the left panel of Figure 3.

(ii) Γ is non-diagonalisable. To treat the non-diagonalisable case, we employ the

above perturbation techniques we developed for the Laplacian, i.e. we approximate

Γ by a diagonalisable matrix Γ. Notice that Γ can be represented in its Jordan form

Γ = QJQ−1, and we can write J = J + E, where E is an ε-perturbation diagonal

matrix as in Proposition 10. The approximation Γ reads as Γ = Q(J + E)Q−1,

and as in Proposition 10, if P denotes the matrix that diagonalises J + E

(i.e. B = P−1(J + E)P is diagonal), then Γ = QPBP−1Q−1. Hence,

ρ ≤ c%κ(QP ) ≤ c%κ(Q)κ(P ) .

By Proposition 10, it is easy to check that

κ(P ) = ‖P‖‖P−1‖ ≤ d

εm−1,

where d > 0 does not depend on ε. The aim is to minimise ρ, which means

minimising κ(P ). The perturbation size ε should be of the same order as β, since

the real parts of the eigenvalues of J+E must be positive. This can be obtained, for

instance, by choosing ε = rβ for some fixed r ∈ (0, 1). This yields to the following

bound

ρ ≤ k

βm−1,

where k is a constant.

Note the different behaviour for the bound as a function of β between the case when

Γ is symmetric and when Γ is non-diagonalisable. This helps to explain the nonlinear

behaviour observed in Figure 1 and in the right panel of Figure 3.

7. Persistence

As in the previous section, we make use of the auxiliary results from Section 5 in

order to prove our main theorem on persistence (Theorem 2), which will be restated for

convenience.


Theorem (persistence). Consider the perturbed network (4) of diffusively coupled

equations fulfilling Assumptions A1–A3, and suppose that

α >ρ

γ

as in Theorem 1. Then there exist δ > 0, C > 0 and εg > 0 such that for all ε0-

perturbations satisfying

‖gi(t, x)‖ ≤ ε0 ≤ εg for all t ∈ R , x ∈ U and i ∈ {1, . . . , n}and initial conditions satisfying ‖xi(t0)− xj(t0)‖ ≤ δ for any i, j ∈ {1, . . . , n}, the

estimate

‖xi(t)− xj(t)‖ ≤ Ce−(αγ−ρ)(t−t0)‖xi(t0)− xj(t0)‖+Cε0

αγ − ρ for all t ≥ t0

holds.

Note that the proof of this theorem does not specifically depend on the fact that the

perturbations gi of the nodes are decoupled; the function G below can depend arbitrarily

on the total state X (or can be subjected to random perturbations).

Proof of Theorem 2. Denote by

G(t,X) = col(g1(t, x1), . . . , gn(t, xn))

the perturbation for the network and note that ‖G‖ ≤ ε0. As in the proof of

Theorem 1, Lemma 13 guarantees that there exists an η-tubular neighbourhood Sηsuch that solutions of the complete system for (s, ξ) cannot escape along s, when εg, η

are sufficiently small.

The perturbed network equation for X = (s, ξ) in Sη now reads as

X = F (t,X)− αL⊗ Γξ +Rh(ξ) +G(t,X) ,

where Rh is the Taylor remainder associated with the coupling function h. Projecting

this equation onto the synchronisation manifold yields an equation for the component

s of X. On the other hand, the differential equation for ξ is given by

ξ = T (t, s(t))ξ +R(t, s(t), ξ) + πN(G(t,1⊗ s+ ξ)) , (42)

see Proposition 7. Let ε(η) denote a Lipschitz constant within Sη of R with respect to

ξ, which does not depend on t.

In the same way as in the proof of Theorem 2, we obtain a variation of constants

formula for solutions of (42),

ξ(t) = Φ(t, t0)ξ(t0) +

∫ t

t0

Φ(t, τ)[R(τ, s(τ), ξ(τ)) + πN(G(τ,1⊗ s(τ) + ξ(τ)))] dτ .

With initial conditions ‖ξ(t0)‖ ≤ ‖πN‖δ, lemmas 8 and 9, and the assumption that

‖ξ(t)‖ ≤ δ1 < η for all t ≥ t0 ,


this leads to the estimate

‖ξ(t)‖ ≤ Ce−µt‖πN‖δ +

∫ t

t0

Ce−µ(t−τ)(ε(δ1)‖ξ(τ)‖+ ‖πN‖ε0) dτ ,

where µ = αγ−ρ. We choose δ < ηC‖πN‖

and δ1, εg sufficiently small and apply Lemma 14

with α = ε(δ1), β = ‖πN‖ε0 to find that

‖ξ(t)‖ ≤ Ceµ(t−t0)‖πN‖(δ − ε0

Cµ

)+C‖πN‖ε0

µfor all t ≥ t0 , (43)

where µ = αγ − ρ − Cε(δ1). As in the proof of Theorem 2, we choose ρ = ρ + Cε(δ1)

instead of ρ and the estimate for ‖xi(t)− xj(t)‖ follows from (43) by adapting δ.

In particular, note that asymptotically, the bound in (43) converges to C‖πN‖ε0αγ−ρ .

Furthermore, it follows from the details of Lemma 8 that the constant C depends on

the Laplacian L only through its conditional number κ(P ).

Finally, we can proof Corollary 3 from the Introduction.

Proof of Corollary 3. This corollary is a direct consequence of our persistence result.

For simplicity, we now endow the space Rnm with the Euclidean norm

‖X‖2 =( n∑

i=1

‖xi‖22

)1/2

for all X = col(x1, . . . , xn) ∈ Rnm .

Note that in view of (43), for large times, we obtain

‖ξ‖2 =

(n∑

i=1

‖s− xi‖22

)1/2

≤ 2Kκ2(P ⊗Q)‖G‖2

µ(44)

where the contraction rate µ is given by µ = αγ − ρ. For simplicity, we omit the

arguments of the functions s, x, G and ξ.

Moreover, κ2(P⊗Q) ≤ κ2(P )κ2(Q), and since the Laplacian is symmetric, it can be

diagonalised by an orthogonal similarity transformation, which implies that κ2(P ) = 1

together with ‖πN‖2 = 1. Moreover, by the equivalence of norms we obtain

‖G‖2 ≤√n‖G‖ ≤ √nε0,

Replacing this estimate in (44) we obtain(

n∑

i=1

‖s− xi‖22

)1/2

≤ K√nε0

µ, (45)

where K = 2Kκ2(Q). We scale equation (45) to obtain(

1

n

n∑

i=1

‖s− xi‖22

)1/2

≤ Kε0

µ, (46)


and applying the sum of squares inequality

1

n

∑

i=1

ai ≤

√√√√ 1

n

n∑

i=1

a2i

leads to

1

n

n∑

i=1

‖s− xi‖2 ≤Kε0

µ. (47)

The triangle inequality implies

1

n

∣∣∣∣∣n∑

i=1

‖s− xj‖2 − ‖xj − xi‖2

∣∣∣∣∣ ≤1

n

n∑

i=1

|‖s− xj‖2 − ‖xj − xi‖2|

≤ 1

n

n∑

i=1

‖s− xi‖2 .

Hence,

1

n

∣∣∣∣∣n∑

i=1

(‖s− xj‖2 − ‖xj − xi‖2)

∣∣∣∣∣ ≤Kε0

µ,

as we control the first sum by (47) we obtain

1

n

n∑

i=1

‖xj − xi‖2 ≤2Kε0

µ.

To conclude the result, we take the sum over the index j and divide by the network size

n. This finishes the proof of this corollary.

8. Generalisations

Although our set-up is very general and includes non-autonomous systems and

non-diagonalisable Laplacians, the assumptions we make are only sufficient for

synchronisation, but not necessary. For instance, let (u, v) = x ∈ R2 and consider

as isolated dynamics x = f(x) with f(x) = (u, u− v), and

x1 = f(x1) + αΓ(x2 − x1)

x2 = f(x2) + αΓ(x1 − x2)with Γ =

(1 0

0 0

).

Note that in this situation Γ has an eigenvalue zero, so Assumption A3 is violated.

However, this coupled system synchronises for α > 1/2. This happens as all instabilities

occurs due to the first variable, and the coupling Γ acts solely on this variable. For a

numerical example of a chaotic system displaying synchronisation with only one variable

coupled, see [24].

The boundedness of the Jacobian D2f in Assumption A1’, and Assumption A3

are used in Lemma 8 to obtain uniform exponential stability of the linear system

(23). For this purpose, we use the diagonal dominance criterion, see (28) in the proof


of Lemma 8. It is clear that one could get uniform exponential stability without

the two above mentioned assumptions. Note that under reasonable assumptions, a

necessary and sufficient condition for uniform exponential stability (and thus persistent

synchronisation) is that the dichotomy spectrum of (23) is contained in the negative

half line [17] (see [11] for a comparative study of numerical methods to approximate the

dichotomy spectrum).

For persistent synchronisation, we thus only require a dichotomy spectrum in

the directions transverse to the synchronisation manifold. Instead we can impose the

stricter condition of normal hyperbolicity (see [10, 14] and e.g. [16] in the context of

synchronisation of networks). That is, we also require that any exponential contraction

tangent to the synchronisation manifold is weaker than in the transverse directions. In

other words, the spectra in the normal and tangential directions must be disjoint and

the normal spectrum must be strictly below the tangential one. In our explicit setup,

this so-called spectral gap condition translates to

ρ− αγ < −r ρ with r ≥ 1.

Under these assumptions we find a stronger form of persistence. Under arbitrary

C1-small perturbations, solutions not only converge into a neighbourhood of the

synchronisation manifold, but an invariant manifold‡S = {xi = hi(s), s ∈ U ⊂ Rm, 1 ≤ i ≤ n}

close to S persists to which these solutions converge. Moreover a stronger ‘shadowing’

or ‘isochrony’ property holds that any solution curve X(t) that converges to S, actually

converges at exponential rate µ to a unique solution curve XS(t) on S in the sense that

there exists a C such that for all t ≥ 0

‖X(t)−XS(t)‖ ≤ Ce−µt ,

with µ close to αγ − ρ.

Acknowledgements. Tiago Pereira was supported by a Marie Curie IIF Fellowship

(Project 303180), Jaap Eldering was supported by the ERC Advanced Grant 267382, and

Martin Rasmussen and Jaap Eldering were supported by an EPSRC Career Acceleration

Fellowship (2010–2015). We also thank CNPq and the Marie Curie IRSES staff exchange

project DynEurBraz.

‡ Both smoothness and uniqueness of this manifold are subtle issues. In general the invariant manifold

cannot be expected to be smoother than Cr. If the synchronisation manifold has a boundary (where it

is only forward invariant), then non-uniqueness follows from local modifications that have to be made

to apply the persistence theorem, see [16]. Note that both results hold, also when the synchronisation

manifold S is noncompact, see [9, Thm 3.1 and Chap. 4].


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